We investigate the mathematical foundations of neural networks in the infinite-width regime through the Neural Tangent Kernel (NTK). We propose the NTK-Eigenvalue-Controlled Residual Network (NTK-ECRN), an architecture integrating Fourier feature embeddings, residual connections with layerwise scaling, and stochastic depth to enable rigorous analysis of kernel evolution during training. Our theoretical contributions include deriving bounds on NTK dynamics, characterizing eigenvalue evolution, and linking spectral properties to generalization and optimization stability. Empirical results on synthetic and benchmark datasets validate the predicted kernel behavior and demonstrate improved training stability and generalization. This work provides a comprehensive framework bridging infinite-width theory and practical deep-learning architectures.

翻译：我们通过神经正切核（NTK）研究了无限宽度体系下神经网络的数学基础。我们提出了NTK-特征值控制残差网络（NTK-ECRN），该架构集成了傅里叶特征嵌入、带层间缩放的残差连接以及随机深度技术，从而能够对训练过程中的核演化进行严格分析。我们的理论贡献包括推导NTK动态的边界、刻画特征值演化过程，并将谱特性与泛化能力及优化稳定性联系起来。在合成数据集和基准数据集上的实证结果验证了预测的核行为，并展示了改进的训练稳定性和泛化性能。本工作为连接无限宽度理论与实用深度学习架构提供了一个综合性的框架。